Renamed some bound variables due to changes in simplifier.
(* Title: ZF/UNITY/UNITY.ML
ID: $Id$
Author: Sidi O Ehmety, Computer Laboratory
Copyright 2001 University of Cambridge
The basic UNITY theory (revised version, based upon the "co" operator)
From Misra, "A Logic for Concurrent Programming", 1994
Proofs ported from HOL
*)
(** SKIP **)
Goal "SKIP:program";
by (auto_tac (claset(), simpset() addsimps
[SKIP_def, program_def, mk_program_def, actionSet_def, cons_absorb]));
qed "SKIP_type";
AddIffs [SKIP_type];
AddTCs [SKIP_type];
(** programify: coersion from anything to program **)
Goalw [programify_def]
"F:program ==> programify(F)=F";
by Auto_tac;
qed "programify_program";
Addsimps [programify_program];
Goalw [programify_def]
"programify(F):program";
by Auto_tac;
qed "programify_in_program";
AddIffs [programify_in_program];
AddTCs [programify_in_program];
(** Collapsing rules: to remove programify from expressions **)
Goalw [programify_def]
"programify(programify(F))=programify(F)";
by Auto_tac;
qed "programify_idem";
Addsimps [programify_idem];
Goal
"Init(programify(F)) = Init(F)";
by (simp_tac (simpset() addsimps [Init_def]) 1);
qed "Init_programify";
Goal
"Acts(programify(F)) = Acts(F)";
by (simp_tac (simpset() addsimps [Acts_def]) 1);
qed "Acts_programify";
Goal
"AllowedActs(programify(F)) = AllowedActs(F)";
by (simp_tac (simpset() addsimps [AllowedActs_def]) 1);
qed "AllowedActs_programify";
Addsimps [Init_programify,Acts_programify,AllowedActs_programify];
(** program inspectors **)
Goal "F:program ==>Id:RawActs(F)";
by (auto_tac (claset(), simpset()
addsimps [program_def, RawActs_def]));
qed "Id_in_RawActs";
Goal "Id:Acts(F)";
by (simp_tac (simpset()
addsimps [Id_in_RawActs, Acts_def]) 1);
qed "Id_in_Acts";
Goal "F:program ==>Id:RawAllowedActs(F)";
by (auto_tac (claset(), simpset()
addsimps [program_def, RawAllowedActs_def]));
qed "Id_in_RawAllowedActs";
Goal "Id:AllowedActs(F)";
by (simp_tac (simpset()
addsimps [Id_in_RawAllowedActs, AllowedActs_def]) 1);
qed "Id_in_AllowedActs";
AddIffs [Id_in_Acts, Id_in_AllowedActs];
AddTCs [Id_in_Acts, Id_in_AllowedActs];
Goal "cons(Id, Acts(F)) = Acts(F)";
by (simp_tac (simpset() addsimps [cons_absorb]) 1);
qed "cons_Id_Acts";
Goal "cons(Id, AllowedActs(F)) = AllowedActs(F)";
by (simp_tac (simpset() addsimps [cons_absorb]) 1);
qed "cons_Id_AllowedActs";
Addsimps [cons_Id_Acts, cons_Id_AllowedActs];
(** inspectors's types **)
Goal
"F:program ==> RawInit(F):condition";
by (auto_tac (claset(), simpset()
addsimps [program_def, RawInit_def]));
qed "RawInit_type";
Goal "Init(F):condition";
by (simp_tac (simpset()
addsimps [RawInit_type, Init_def]) 1);
qed "Init_type";
Goal
"F:program ==> RawActs(F):actionSet";
by (auto_tac (claset(), simpset()
addsimps [program_def, RawActs_def, actionSet_def]));
qed "RawActs_type";
Goal
"Acts(F):actionSet";
by (simp_tac (simpset()
addsimps [RawActs_type, Acts_def]) 1);
qed "Acts_type";
Goal
"F:program ==> RawAllowedActs(F):actionSet";
by (auto_tac (claset(), simpset()
addsimps [program_def, RawAllowedActs_def, actionSet_def]));
qed "RawAllowedActs_type";
Goal
"AllowedActs(F): actionSet";
by (simp_tac (simpset()
addsimps [RawAllowedActs_type, AllowedActs_def]) 1);
qed "AllowedActs_type";
AddIffs [Init_type, Acts_type, AllowedActs_type];
AddTCs [Init_type, Acts_type, AllowedActs_type];
Goal "x:Init(F) ==> x:state";
by (cut_inst_tac [("F", "F")] Init_type 1);
by (auto_tac (claset(), simpset() addsimps [condition_def]));
qed "InitD";
Goal "act:Acts(F) ==> act<=state*state";
by (cut_inst_tac [("F", "F")] Acts_type 1);
by (rewrite_goals_tac [actionSet_def]);
by Auto_tac;
qed "ActsD";
Goal "act:AllowedActs(F) ==> act<=state*state";
by (cut_inst_tac [("F", "F")] AllowedActs_type 1);
by (rewrite_goals_tac [actionSet_def]);
by Auto_tac;
qed "AllowedActsD";
(** More simplification rules involving state
and Init, Acts, and AllowedActs **)
Goal "state <= Init(F) <-> Init(F)=state";
by (cut_inst_tac [("F", "F")] Init_type 1);
by (auto_tac (claset(),
simpset() addsimps [condition_def]));
qed "Init_state_eq";
Goal "action <= Acts(F) <-> Acts(F)=action";
by (cut_inst_tac [("F", "F")] Acts_type 1);
by (auto_tac (claset(),
simpset() addsimps [actionSet_def]));
qed "Acts_action_eq";
Goal "action <= AllowedActs(F) <-> AllowedActs(F)=action";
by (cut_inst_tac [("F", "F")] AllowedActs_type 1);
by (auto_tac (claset(),
simpset() addsimps [actionSet_def]));
qed "AllowedActs_action_eq";
(** Eliminating `Int state' from expressions **)
Goal
"Init(F) Int state = Init(F)";
by (cut_inst_tac [("F", "F")] Init_type 1);
by (auto_tac (claset(),
simpset() addsimps [condition_def]));
qed "Init_Int_state";
Goal
"state Int Init(F) = Init(F)";
by (stac (Int_commute) 1);
by (simp_tac (simpset()
addsimps [Init_Int_state]) 1);
qed "Init_Int_state2";
Goal
"Acts(F) Int action = Acts(F)";
by (cut_inst_tac [("F", "F")] Acts_type 1);
by (auto_tac (claset(),
simpset() addsimps [actionSet_def]));
qed "Acts_Int_action";
Goal
"action Int Acts(F) = Acts(F)";
by (simp_tac (simpset() addsimps
[Int_commute, Acts_Int_action]) 1);
qed "Acts_Int_action2";
Goal
"AllowedActs(F) Int action = AllowedActs(F)";
by (cut_inst_tac [("F", "F")] AllowedActs_type 1);
by (auto_tac (claset(),
simpset() addsimps [actionSet_def]));
qed "AllowedActs_Int_action";
Goal
"action Int AllowedActs(F) = AllowedActs(F)";
by (simp_tac (simpset() addsimps
[Int_commute, AllowedActs_Int_action]) 1);
qed "AllowedActs_Int_action2";
Addsimps
[Init_state_eq, Acts_action_eq, AllowedActs_action_eq,
Init_Int_state, Init_Int_state2, Acts_Int_action, Acts_Int_action2,
AllowedActs_Int_action,AllowedActs_Int_action2];
(** mk_program **)
Goal "mk_program(init, acts, allowed):program";
by (auto_tac (claset(), simpset()
addsimps [program_def, mk_program_def, condition_def, actionSet_def]));
qed "mk_program_type";
AddIffs [mk_program_type];
AddTCs [mk_program_type];
Goal "RawInit(mk_program(init, acts, allowed)) = init Int state";
by (auto_tac (claset(), simpset()
addsimps [RawInit_def, mk_program_def]));
qed "RawInit_eq";
Goal "Init(mk_program(init, acts, allowed)) = init Int state";
by (auto_tac (claset(), simpset()
addsimps [Init_def, RawInit_eq]));
qed "Init_eq";
Addsimps [Init_eq];
Goalw [RawActs_def]
"RawActs(mk_program(init, acts, allowed)) \
\ = cons(Id, acts Int Pow(state*state))";
by (auto_tac (claset(), simpset()
addsimps [mk_program_def]));
qed "RawActs_eq";
Goalw [Acts_def]
"Acts(mk_program(init, acts, allowed)) \
\ = cons(Id, acts Int Pow(state*state))";
by (auto_tac (claset(),
simpset() addsimps [RawActs_eq]));
qed "Acts_eq";
Addsimps [Acts_eq];
Goalw [RawAllowedActs_def]
"RawAllowedActs(mk_program(init, acts, allowed)) \
\ = cons(Id, allowed Int action)";
by (auto_tac (claset(),
simpset() addsimps [mk_program_def]));
qed "RawAllowedActs_eq";
Goalw [AllowedActs_def]
"AllowedActs(mk_program(init, acts, allowed)) \
\ = cons(Id, allowed Int action)";
by (auto_tac (claset(),
simpset() addsimps [RawAllowedActs_eq]));
qed "AllowedActs_eq";
Addsimps [AllowedActs_eq];
(**Init, Acts, and AlowedActs of SKIP **)
Goal "RawInit(SKIP) = state";
by (auto_tac (claset(), simpset()
addsimps [SKIP_def, RawInit_eq]));
qed "RawInit_SKIP";
Goal "Init(SKIP) = state";
by (simp_tac (simpset()
addsimps [Init_def, RawInit_SKIP]) 1);
qed "Init_SKIP";
Addsimps [Init_SKIP];
Goal "RawActs(SKIP) = {Id}";
by (auto_tac (claset(), simpset()
addsimps [SKIP_def, RawActs_eq]));
qed "RawActs_SKIP";
Goal "Acts(SKIP) = {Id}";
by (simp_tac (simpset()
addsimps [Acts_def, RawActs_SKIP]) 1);
qed "Acts_SKIP";
Addsimps [Acts_SKIP];
Goal "RawAllowedActs(SKIP) = action";
by (auto_tac (claset(), simpset()
addsimps [SKIP_def, RawAllowedActs_eq]));
qed "RawAllowedActs_SKIP";
Goal "AllowedActs(SKIP) = action";
by (simp_tac (simpset()
addsimps [AllowedActs_def, RawAllowedActs_SKIP]) 1);
qed "AllowedActs_SKIP";
Addsimps [AllowedActs_SKIP];
(** Equality for UNITY programs **)
Goal
"F:program ==> mk_program(RawInit(F), RawActs(F), RawAllowedActs(F))=F";
by (auto_tac (claset(), simpset() addsimps
[program_def, mk_program_def, RawInit_def,
RawActs_def, RawAllowedActs_def, actionSet_def, condition_def]));
by (REPEAT(Blast_tac 1));
qed "raw_surjective_mk_program";
Goal
"mk_program(Init(F), Acts(F), AllowedActs(F)) = programify(F)";
by (auto_tac (claset(), simpset() addsimps
[Init_def, Acts_def, AllowedActs_def,
raw_surjective_mk_program]));
qed "surjective_mk_program";
Goal "[|Init(F) = Init(G); Acts(F) = Acts(G); \
\ AllowedActs(F) = AllowedActs(G); F:program; G:program |] ==> F = G";
by (stac (programify_program RS sym) 1);
by (rtac sym 2);
by (stac (programify_program RS sym) 2);
by (stac (surjective_mk_program RS sym) 3);
by (stac (surjective_mk_program RS sym) 3);
by (ALLGOALS(Asm_simp_tac));
qed "program_equalityI";
val [major,minor] =
Goal "[| F = G; \
\ [| Init(F) = Init(G); Acts(F) = Acts(G); AllowedActs(F) = AllowedActs(G) |]\
\ ==> P |] ==> P";
by (rtac minor 1);
by (auto_tac (claset(), simpset() addsimps [major]));
qed "program_equalityE";
Goal "[| F:program; G:program |] ==>(F=G) <-> \
\ (Init(F) = Init(G) & Acts(F) = Acts(G) & AllowedActs(F) = AllowedActs(G))";
by (blast_tac (claset() addIs [program_equalityI, program_equalityE]) 1);
qed "program_equality_iff";
Addsimps [surjective_mk_program];
(*** These rules allow "lazy" definition expansion
...skipping 1 line
***)
Goal "F == mk_program (init,acts,allowed) ==> Init(F) = init Int state";
by Auto_tac;
qed "def_prg_Init";
Goal "F == mk_program (init,acts,allowed) ==> Acts(F) = cons(Id, acts Int action)";
by Auto_tac;
qed "def_prg_Acts";
Goal "F == mk_program (init,acts,allowed) ==> \
\ AllowedActs(F) = cons(Id, allowed Int action)";
by Auto_tac;
qed "def_prg_AllowedActs";
val [rew] = goal thy
"[| F == mk_program (init,acts,allowed) |] \
\ ==> Init(F) = init Int state & Acts(F) = cons(Id, acts Int action)";
by (rewtac rew);
by Auto_tac;
qed "def_prg_simps";
(*An action is expanded only if a pair of states is being tested against it*)
Goal "[| act == {<s,s'>:A*B. P(s, s')} |] ==> \
\ (<s,s'>:act) <-> (<s,s'>:A*B & P(s, s'))";
by Auto_tac;
qed "def_act_simp";
fun simp_of_act def = def RS def_act_simp;
(*A set is expanded only if an element is being tested against it*)
Goal "A == B ==> (x : A) <-> (x : B)";
by Auto_tac;
qed "def_set_simp";
fun simp_of_set def = def RS def_set_simp;
(*** co ***)
val prems = Goalw [constrains_def]
"[|(!!act s s'. [| act: Acts(F); <s,s'>:act; s:A|] ==> s':A'); \
\ F:program; A:condition; A':condition |] ==> F:A co A'";
by (blast_tac(claset() addIs prems) 1);
qed "constrainsI";
Goalw [constrains_def]
"F:A co B ==> ALL act:Acts(F). act``A<=B";
by (Blast_tac 1);
qed "constrainsD";
Goalw [constrains_def]
"F:A co B ==> F:program & A:condition & B:condition";
by (Blast_tac 1);
qed "constrainsD2";
Goalw [constrains_def]
"[| F:program; B:condition |] ==> F : 0 co B";
by (Blast_tac 1);
qed "constrains_empty";
Goalw [constrains_def]
"[| F:program; A:condition |] ==>(F : A co 0) <-> (A=0)";
by (auto_tac (claset() addSDs [bspec],
simpset() addsimps [condition_def]));
qed "constrains_empty2";
Goalw [constrains_def]
"[| F:program; B:condition |] ==> (F: state co B) <-> (B = state)";
by (auto_tac (claset() addSDs [bspec] addDs [ActsD],
simpset() addsimps [condition_def]));
qed "constrains_state";
Goalw [constrains_def]
"[| F:program; A:condition |] ==> F : A co state";
by (auto_tac (claset() addDs [ActsD], simpset()));
qed "constrains_state2";
Addsimps [constrains_empty, constrains_empty2,
constrains_state, constrains_state2];
(*monotonic in 2nd argument*)
Goalw [constrains_def]
"[| F:A co A'; A'<=B'; B':condition |] ==> F : A co B'";
by (Blast_tac 1);
qed "constrains_weaken_R";
(*anti-monotonic in 1st argument*)
Goalw [constrains_def, condition_def]
"[| F : A co A'; B<=A |] ==> F : B co A'";
by (Blast_tac 1);
qed "constrains_weaken_L";
Goal
"[| F : A co A'; B<=A; A'<=B'; B':condition |] ==> F : B co B'";
by (dtac constrains_weaken_R 1);
by (dtac constrains_weaken_L 3);
by (REPEAT(Blast_tac 1));
qed "constrains_weaken";
(** Union **)
Goalw [constrains_def]
"[| F : A co A'; F:B co B' |] ==> F:(A Un B) co (A' Un B')";
by (Asm_full_simp_tac 1);
by (Blast_tac 1);
qed "constrains_Un";
Goalw [constrains_def]
"[| F : A co A'; F:B co B' |] ==> F:(A Un B) co (A' Un B')";
by Auto_tac;
by (asm_full_simp_tac
(simpset() addsimps [condition_def]) 1);
by (Blast_tac 1);
qed "constrains_Un";
Goalw [constrains_def]
"[| ALL i:I. F:A(i) co A'(i); F:program |] ==> \
\ F:(UN i:I. A(i)) co (UN i:I. A'(i))";
by (Clarify_tac 1);
by (Blast_tac 1);
bind_thm ("constrains_UN", ballI RS result());
Goalw [constrains_def]
"(A Un B) co C = (A co C) Int (B co C)";
by (rtac equalityI 1);
by (ALLGOALS(Asm_full_simp_tac));
by (ALLGOALS(Blast_tac));
qed "constrains_Un_distrib";
Goalw [constrains_def]
"i:I ==> (UN i:I. A(i)) co B = (INT i:I. A(i) co B)";
by (rtac equalityI 1);
by Safe_tac;
by (ALLGOALS(Asm_full_simp_tac));
by (ALLGOALS(Blast_tac));
qed "constrains_UN_distrib";
Goalw [constrains_def]
"[| A:condition; B:condition |] \
\ ==> C co (A Int B) = (C co A) Int (C co B)";
by (rtac equalityI 1);
by (ALLGOALS(Clarify_tac));
by (ALLGOALS(Blast_tac));
qed "constrains_Int_distrib";
Goalw [constrains_def] "[| i:I; ALL i:I. B(i):condition |] ==> \
\ A co (INT i:I. B(i)) = (INT i:I. A co B(i))";
by (rtac equalityI 1);
by Safe_tac;
by (ALLGOALS(Blast_tac));
qed "constrains_INT_distrib";
(** Intersection **)
Goalw [constrains_def]
"[| F : A co A'; F : B co B' |] ==> F : (A Int B) co (A' Int B')";
by (Clarify_tac 1);
by (Blast_tac 1);
qed "constrains_Int";
Goalw [constrains_def]
"[| ALL i:I. F : A(i) co A'(i); F:program |] \
\ ==> F : (INT i:I. A(i)) co (INT i:I. A'(i))";
by (case_tac "I=0" 1);
by (asm_full_simp_tac (simpset() addsimps [Inter_def]) 1);
by (etac not_emptyE 1);
by Safe_tac;
by (dres_inst_tac [("x", "xd")] bspec 1);
by (ALLGOALS(Blast_tac));
bind_thm ("constrains_INT", ballI RS result());
(* This rule simulates the HOL's one for (INT z. A i) co (INT z. B i) *)
Goalw [constrains_def, condition_def]
"[| ALL z. F: {s:state. P(s, z)} co {s:state. Q(s, z)}; F:program |] ==> \
\ F: {s:state. ALL z. P(s, z)} co {s:state. ALL z. Q(s, z)}";
by (auto_tac (claset() addSDs [bspec] addDs [ActsD],
simpset() addsimps [condition_def]));
by (Blast_tac 1);
bind_thm ("constrains_All", allI RS result());
Goalw [constrains_def]
"F : A co A' ==> A <= A'";
by (auto_tac (claset() addSDs [bspec], simpset()));
qed "constrains_imp_subset";
(*The reasoning is by subsets since "co" refers to single actions
only. So this rule isn't that useful.*)
Goalw [constrains_def]
"[| F : A co B; F : B co C |] ==> F : A co C";
by Auto_tac;
by (dres_inst_tac [("x", "act")] bspec 1);
by (dres_inst_tac [("x", "Id")] bspec 2);
by (auto_tac (claset() addDs [ActsD],
simpset() addsimps [condition_def]));
qed "constrains_trans";
Goalw [constrains_def]
"[| F : A co (A' Un B); F : B co B' |] ==> F : A co (A' Un B')";
by Auto_tac;
by (dres_inst_tac [("x", "Id")] bspec 1);
by (dres_inst_tac [("x", "act")] bspec 2);
by (auto_tac (claset(), simpset() addsimps [condition_def]));
qed "constrains_cancel";
(*** unless ***)
Goalw [unless_def] "F : (A-B) co (A Un B) ==> F : A unless B";
by (assume_tac 1);
qed "unlessI";
Goalw [unless_def] "F :A unless B ==> F : (A-B) co (A Un B)";
by (assume_tac 1);
qed "unlessD";
Goalw [unless_def, constrains_def]
"F: A unless B ==> F:program & A:condition & B:condition";
by Auto_tac;
qed "unlessD2";
(*** initially ***)
Goalw [initially_def]
"[| Init(F)<=A; F:program; A:condition |] ==> F:initially(A)";
by (Blast_tac 1);
qed "initiallyI";
Goalw [initially_def]
"F:initially(A) ==> Init(F)<=A";
by (Blast_tac 1);
qed "initiallyD";
Goalw [initially_def]
"F:initially(A) ==> F:program & A:condition";
by (Blast_tac 1);
qed "initiallyD2";
(*** stable ***)
Goalw [stable_def]
"F : A co A ==> F : stable(A)";
by (assume_tac 1);
qed "stableI";
Goalw [stable_def] "F : stable(A) ==> F : A co A";
by (assume_tac 1);
qed "stableD";
Goalw [stable_def, constrains_def]
"F:stable(A)==> F:program & A:condition";
by (Blast_tac 1);
qed "stableD2";
Goalw [stable_def, constrains_def]
"stable(state) = program";
by (auto_tac (claset() addDs [ActsD], simpset()));
qed "stable_state";
Addsimps [stable_state];
(** Union **)
Goalw [stable_def]
"[| F : stable(A); F : stable(A') |] ==> F : stable(A Un A')";
by (blast_tac (claset() addIs [constrains_Un]) 1);
qed "stable_Un";
val [major, minor] = Goalw [stable_def]
"[| (!!i. i:I ==> F : stable(A(i))); F:program |] ==> F:stable (UN i:I. A(i))";
by (cut_facts_tac [minor] 1);
by (blast_tac (claset() addIs [constrains_UN, major]) 1);
qed "stable_UN";
Goalw [stable_def]
"[| F : stable(A); F : stable(A') |] ==> F : stable (A Int A')";
by (blast_tac (claset() addIs [constrains_Int]) 1);
qed "stable_Int";
val [major, minor] = Goalw [stable_def]
"[| (!!i. i:I ==> F : stable(A(i))); F:program |] \
\ ==> F : stable (INT i:I. A(i))";
by (cut_facts_tac [minor] 1);
by (blast_tac (claset() addIs [constrains_INT, major]) 1);
qed "stable_INT";
Goalw [stable_def]
"[| ALL z. F: stable({s:state. P(s, z)}); F:program |] ==> \
\ F: stable({s:state. ALL z. P(s, z)})";
by (rtac constrains_All 1);
by Auto_tac;
bind_thm("stable_All", allI RS result());
Goalw [stable_def, constrains_def]
"[| F : stable(C); F : A co (C Un A') |] ==> F : (C Un A) co (C Un A')";
by (Clarify_tac 1);
by (Blast_tac 1);
qed "stable_constrains_Un";
Goalw [stable_def, constrains_def]
"[| F : stable(C); F : (C Int A) co A' |] ==> F : (C Int A) co (C Int A')";
by (Clarify_tac 1);
by (Blast_tac 1);
qed "stable_constrains_Int";
(*[| F : stable(C); F : (C Int A) co A |] ==> F : stable (C Int A) *)
bind_thm ("stable_constrains_stable", stable_constrains_Int RS stableI);
(** invariant **)
Goalw [invariant_def, initially_def]
"invariant(A) = initially(A) Int stable(A)";
by (blast_tac (claset() addDs [stableD2]) 1);
qed "invariant_eq";
val invariant_def2 = invariant_eq RS eq_reflection;
Goalw [invariant_def]
"[| Init(F)<=A; F:stable(A) |] ==> F : invariant(A)";
by (blast_tac (claset() addDs [stableD2]) 1);
qed "invariantI";
Goalw [invariant_def]
"F:invariant(A) ==> Init(F)<=A & F:stable(A)";
by Auto_tac;
qed "invariantD";
Goalw [invariant_def]
"F:invariant(A) ==> F:program & A:condition";
by (blast_tac (claset() addDs [stableD2]) 1);
qed "invariantD2";
(*Could also say "invariant A Int invariant B <= invariant (A Int B)"*)
Goalw [invariant_def]
"[| F : invariant(A); F : invariant(B) |] ==> F : invariant(A Int B)";
by (asm_full_simp_tac (simpset() addsimps [stable_Int]) 1);
by (Blast_tac 1);
qed "invariant_Int";
(** The Elimination Theorem. The "free" m has become universally quantified!
Should the premise be !!m instead of ALL m ? Would make it harder to use
in forward proof. **)
Goalw [constrains_def]
"[| ALL m:M. F : {s:S. x(s) = m} co B(m); F:program |] \
\ ==> F:{s:S. x(s):M} co (UN m:M. B(m))";
by Safe_tac;
by (Blast_tac 1);
by (auto_tac (claset(), simpset() addsimps [condition_def]));
qed "elimination";
(* As above, but for the special case of S=state *)
Goal "[| ALL m:M. F : {s:state. x(s) = m} co B(m); F:program |] \
\ ==> F:{s:state. x(s):M} co (UN m:M. B(m))";
by (rtac elimination 1);
by (ALLGOALS(Clarify_tac));
qed "eliminiation2";
Goalw [constrains_def, strongest_rhs_def]
"[| F:program; A:condition |] ==>F : A co (strongest_rhs(F,A))";
by (auto_tac (claset() addDs [ActsD], simpset()));
qed "constrains_strongest_rhs";
Goalw [constrains_def, strongest_rhs_def]
"F : A co B ==> strongest_rhs(F,A) <=B";
by Safe_tac;
by (dtac InterD 1);
by (auto_tac (claset(),
simpset() addsimps [condition_def]));
qed "strongest_rhs_is_strongest";
(*** increasing ***)
Goalw [increasing_on_def]
"[| F:increasing[A](f, r); z:A |] ==> F:stable({s:state. <z, f`s>:r})";
by (Blast_tac 1);
qed "increasing_onD";
Goalw [increasing_on_def]
"F:increasing[A](f, r) ==> F:program & f:state->A & part_order(A,r)";
by (auto_tac (claset(), simpset() addsimps [INT_iff]));
qed "increasing_onD2";
Goalw [increasing_on_def, stable_def]
"[| part_order(A,r); c:A; F:program |] ==> F : increasing[A](lam s:state. c, r)";
by (auto_tac (claset(), simpset() addsimps [INT_iff]));
by (force_tac (claset() addSDs [bspec, ActsD],
simpset() addsimps [constrains_def, condition_def]) 1);
qed "increasing_on_constant";
Addsimps [increasing_on_constant];
Goalw [increasing_on_def, stable_def, constrains_def, part_order_def]
"!!f. g:mono_map(A,r,A,r) \
\ ==> increasing[A](f, r) <= increasing[A](g O f, r)";
by (asm_full_simp_tac (simpset() addsimps [INT_iff,condition_def, mono_map_def]) 1);
by (auto_tac (claset() addIs [comp_fun], simpset() addsimps [mono_map_def]));
by (force_tac (claset() addSDs [bspec, ActsD], simpset()) 1);
by (subgoal_tac "xc:state" 1);
by (force_tac (claset() addSDs [ActsD], simpset()) 2);
by (subgoal_tac "f`xd:A & f`xc:A" 1);
by (blast_tac (claset() addDs [apply_type]) 2);
by (rotate_tac 3 1);
by (dres_inst_tac [("x", "f`xd")] bspec 1);
by (Asm_simp_tac 1);
by (REPEAT(etac conjE 1));
by (rotate_tac ~2 1);
by (dres_inst_tac [("x", "act")] bspec 1);
by (Asm_simp_tac 1);
by (dres_inst_tac [("c", "xc")] subsetD 1);
by (rtac imageI 1);
by Auto_tac;
by (asm_full_simp_tac (simpset() addsimps [refl_def]) 1);
by (dres_inst_tac [("x", "f`xd")] bspec 1);
by (dres_inst_tac [("x", "f`xc")] bspec 2);
by (ALLGOALS(Asm_simp_tac));
by (dres_inst_tac [("b", "g`(f`xd)")] trans_onD 1);
by Auto_tac;
qed "mono_increasing_on_comp";
(*Holds by the theorem (succ(m) le n) = (m < n) *)
Goalw [increasing_on_def, nat_order_def]
"[| F:increasing[nat](f, nat_order); z:nat |] \
\ ==> F: stable({s:state. z < f`s})";
by (Clarify_tac 1);
by (asm_full_simp_tac (simpset() addsimps [INT_iff]) 1);
by Safe_tac;
by (dres_inst_tac [("x", "succ(z)")] bspec 1);
by (auto_tac (claset(), simpset() addsimps [apply_type, Collect_conj_eq]));
by (subgoal_tac "{x: state . f ` x : nat} = state" 1);
by Auto_tac;
qed "strict_increasing_onD";